10 I ELEMENTS OF THE THEORY OF SETS

If both F and G are injective (resp. surjective, bijective), then H = G o F
is injective (resp. surjective, bijective); if F and G are bijections, then
H-i = p-i c G"1. If F is a bijection, then F"1 ° F is the identity mapping
of X, and F ° F'1 the identity mapping of Y.

Conversely, if F is a mapping of X into Y, G a mapping of Y into X
such that G ° F = lx and F ° G = 1Y, F and G are bijections inverse to each
other, for the first relation implies that F is injective and G surjective, and
the second that G is injective and F surjective.

Let T be a set, Ft a mapping of X into Y, F2 a mapping of Y into Z, F3
a mapping of Z into T. Then F3 o (F2 * FJ = (F3 o F2) ° ¥1 by definition;
it is a mapping of X into T, also written F3» F2 ° F!. Composition of any
finite number of mappings is defined in the same way.

PROBLEMS

1. Let A, B, C, D be sets,/a mapping of A into B, g a mapping of B into C, h a mapping
of C into D. Show that if g o/and h o g are bijective, /, g, h are all bijective.

2. Let A, B, C be sets,/a mapping of A into B, g a mapping of B into C, h a mapping
of C into A. Show that if, among the mappings h °g o/, g o/o h, /° h °g, two are
surjective and the third injective, or two are injective and the third surjective, then
all three mappings /, g, h are bijective.

3. Let F be a subset of X x Y, G a subset of Y x X. With the notations of Problem 3 of
Section 1.5, suppose that for any x e X, G(F(x)) = {x} and for any y e Y, F(G(y)) - {y}*
Show that F is the graph of a bijection of X onto Y and G the graph of the inverse of F.

4. Let X, Y be two sets,/an injection of X into Y, g an injection of Y into X. Show that
there exist two subsets A, B of X such that B = X — A, two subsets A', B' of Y such
that B' = Y - A', and that A' =/(A) and B = g(W). [Let R = X - #(Y), and h = g ° /;
take for A the intersection of all subsets M of X such that M "=> R u h(M).]

8. FAMILIES OF ELEMENTS. UNION, INTERSECTION, AND PRODUCTS
OF FAMILIES OF SETS. EQUIVALENCE RELATIONS

Let L and X be two sets. A mapping of L into X is sometimes also
called a family of elements 0/X, having L as set of indices, and it is written
A-+XI, or fe)AeL, or simply (*A) when no confusion can arise. The most
important examples are given by sequences (finite or infinite) which cor-
respond to the cases in which L is a finite or infinite subset of the set N of
integers ^0.

Care must be taken to distinguish a family (xA)AeL of elements of X
from the subset of X whose elements are the elements of the family, which
is the image of L by the mapping A-*^, and can very well consist only
of one element; different families may thus have the same set of elements.it possible for me to teach this course along the lines I had planned